On differences of perfect powers and prime powers

Note on the rank of quadratic twists of Mordell equations

This paper is devoted to the study of a binary inequality that links prime numbers with square-free integers. The main goal is to show that, for any exponent lying in a certain interval greater than one and smaller than a critical constant, and for every sufficiently large positive number $$N$$ , the inequality under investigation admits a solution. In such a solution one of the variables is a prime number, while the other is restricted to be square-free. The approach relies on analytic number theory, in particular on estimates for exponential sums involving prime numbers together with sieve methods to control the contribution of square-free integers. The obtained result demonstrates that even under the arithmetic restriction of being square-free, the inequality can still be solved for infinitely many sufficiently large values of $$N$$ . This extends previous research, where either primes or unrestricted integers were considered, and provides a new contribution to the understanding of additive problems involving primes and special classes of integers.

Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves

Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves

This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct bounds for linear forms in the logarithms of algebraic numbers, and the necessary results are stated without proof at the end of the chapter. *** DIRECT SUPPORT *** A00I6B17 00002

In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and p-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

The power Fibonacci sequence in $\mathbb{Z}_m[\sqrt{\delta}]$ is defined as a Fibonacci sequence \linebreak $F_n = F_{n-1} + F_{n-2}$ where $F_0 = 1$ and $F_1 = a $, such that $a\in \mathbb{Z}_m[\sqrt{\delta}]$ and $F_n\equiv a^n \:(\mathrm{mod}\: m)$, for all $n\in \mathbb{N}\cup \{0\}$. In this paper, we investigated the existence of power Fibonacci sequences in $\mathbb{Z}_m[\sqrt{\delta}]$, and the number of power Fibonacci sequences in $\mathbb{Z}_m[\sqrt{\delta}]$ for a given $m$, where $\delta$ is a square-free integer. Furthermore, we determined explicitly all power Fibonacci sequences in $\mathbb{Z}_{p^k}[\sqrt{\delta}]$, where $p$ is a prime number.

We solve the equation xa + xb + 1 = yq in positive integers x, y, a, b and q with a > b and q ≥ 2 coprime to φ(x). This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and p-adic logarithms, the hypergeometric method of Thue and Siegel applied p-adically, local methods, and the algorithmic resolution of Thue equations.

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

In this research we are going to define two new concepts: a) "The Potential of Events" (EP) and b) "The Catholic Information" (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions ( CLP ) which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function (s) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, , 9.

Explicit integral basis of [formula omitted

A Fourier-like transform is defined over a ring of quadratic integers modulo a prime number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> in the quadratic field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(\sqrt{m})</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is a square-free integer. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> is a Fermat prime, one can utilize the fast Fourier transform (FFT) algorithm over the resulting finite fields to yield fast convolutions of quadratic integer sequences in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(\sqrt{m})</tex> . The theory is also extended to a direct sum of such finite fields. From these results, it is shown that Fourier-like transforms can also be defined over the quadratic integers in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R( \sqrt{m})</tex> modulo a nonprime Fermat number.

Let 1∕2≤β<1, p be a generic prime number and fβ be a random multiplicative function supported on the squarefree integers such that (fβ(p))p is an i.i.d. sequence of random variables with distribution P(f(p)=−1)=β=1−P(f(p)=+1). Let Fβ be the Dirichlet series of fβ. We prove a formula involving measure-preserving transformations that relates the Riemann ζ function with the Dirichlet series of Fβ, for certain values of β, and give an application. Further, we prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted partial sum of fβ.

Let K = Q ( $\sqrt{m}$) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < | s| < r and gcd( r,s) = 1. A positive integer n is called a ( K,θ)-congruent number if there is a triangle, called the ( K,θ, n)-triangles, with sides in K having θ as an angle and nα θ as area, where α θ = $\sqrt{r^2-s^2}$. Consider the ( K,θ)-congruent number elliptic curve E n,θ : y 2 = x( x + ( r + s) n) ( x − ( r − s) n) defined over K. Denote the squarefree part of positive integer t by sqf( t). In this work, it is proved that if m ≠ sqf(2 r( r − s)) and mn ≠ 2, 3, 6, then n is a ( K,θ)-congruent number if and only if the Mordell-Weil group E n,θ ( K) has positive rank, and all of the ( K, θ, n)-triangles are classified in four types.

The complex formation in chloroform solution of poly(N‐vinylcarbazole) and N‐ethylcarbazole with the electron acceptors tetracyanoethylene, tetrachloro‐p‐benzoquinone (chloranil) and 1,3,5‐trinitrobenzene was studied. The stability of the donor‐acceptor complexes with mole ratios of 1:1 was determined spectrophotometrically by a modified method of successive dilution.A mathematical method was developed for a simultaneous determination of the extinction coefficient and of the stability constant by conversion of a quadratic relation into a linear form and by application of the least square method. The experimental results were treated by a computer program.The values obtained by this method for the extinction coefficients and for the stability constants of N‐ethyl carbazole complexes are higher than those of poly(N‐vinylcarbazole) complexes. The change of the electron affinity of the acceptors does not affect considerably the stability constants of the complexes.